Variational Multiscale Research Articles

A residual-based variational multiscale method with weak imposition of boundary conditions for buoyancy-driven flows

There is growing interest in high fidelity simulations of buoyancy-driven flows in indoor environments. This is driven to a large extent by the push to utilize naturally occurring flows to reduce energy consumption in the built environment. Naturally occurring (buoyancy-driven) flows exhibit spatiotemporal thermal fluctuations, and accurately capturing these thermal variations is essential to calculate energy usage. In this work, we deploy a residual-based variational multiscale (VMS) finite element large-eddy simulation model for accurately modeling buoyancy-driven flows in enclosed environments. We use the canonical example of the Rayleigh–Bénard convection problem in 2D and 3D to verify and validate the VMS model. We show good comparison with benchmark numerical and experimental results across seven orders of magnitude variation in Rayleigh numbers (Ra∼103 to 1010), covering laminar, transition, and turbulent regimes without any extra treatments. We additionally employ weak enforcement of Dirichlet boundary conditions for both velocity and temperature, and show that comparable results can be produced with much coarser meshes. This confirms that the VMS framework with weak imposition of Dirichlet boundary conditions is a computationally efficient approach to model buoyancy-driven flow physics in complex indoor environments.

Computational analysis methods for complex unsteady flow problems

In this lead paper of the special issue, we provide a brief summary of the stabilized and multiscale methods in fluid dynamics. We highlight the key features of the stabilized and multiscale scale methods, and variational methods in general, that make these approaches well suited for computational analysis of complex, unsteady flows encountered in modern science and engineering applications. We mainly focus on the recent developments. We discuss application of the variational multiscale (VMS) methods to fluid dynamics problems involving computational challenges associated with high-Reynolds-number flows, wall-bounded turbulent flows, flows on moving domains including subdomains in relative motion, fluid–structure interaction (FSI), and complex-fluid flows with FSI.

Space–time Isogeometric flow analysis with built-in Reynolds-equation limit

We present a space–time (ST) computational flow analysis method with built-in Reynolds-equation limit. The method enables solution of lubrication fluid dynamics problems with a computational cost comparable to that of the Reynolds-equation model for the comparable solution quality, but with the computational flexibility to go beyond the limitations of the Reynolds-equation model. The key components of the method are the ST Variational Multiscale (ST-VMS) method, ST Isogeometric Analysis (ST-IGA), and the ST Slip Interface (ST-SI) method. The VMS feature of the ST-VMS serves as a numerical stabilization method with a good track record, the moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces, and the higher-order accuracy of the ST framework strengthens both features. The ST-IGA enables more accurate representation of the solid-surface geometries and increased accuracy in the flow solution in general. With the ST-IGA, even with just one quadratic NURBS element across the gap of the lubrication fluid dynamics problem, we reach a solution quality comparable to that of the Reynolds-equation model. The ST-SI enables moving-mesh computation when the spinning solid surface is noncircular. The mesh covering the solid surface spins with it, retaining the high-resolution representation of the flow near the surface, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. We present detailed 2D test computations to show how the method performs compared to the Reynolds-equation model, compared to finite element discretization, at different circumferential and normal mesh refinement levels, when there is an SI in the mesh, and when the no-slip boundary conditions are weakly-enforced.

Methods for computation of flow-driven string dynamics in a pump and residence time

We present methods for computation of flow-driven string dynamics in a pump and related residence time. The string dynamics computations help us understand how the strings carried by a fluid interact with the pump surfaces, including the blades, and get stuck on or around those surfaces. The residence time computations help us to have a simplified but quick understanding of the string behavior. The core computational method is the Space–Time Variational Multiscale (ST-VMS) method, and the other key methods are the ST Isogeometric Analysis (ST-IGA), ST Slip Interface (ST-SI) method, ST/NURBS Mesh Update Method (STNMUM), a general-purpose NURBS mesh generation method for complex geometries, and a one-way-dependence model for the string dynamics. The ST-IGA with NURBS basis functions in space is used in both fluid mechanics and string structural dynamics. The ST framework provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the turbulent nature of the unsteady flow, and the moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the pump geometry and increased accuracy in the flow solution. The IGA discretization also enables increased accuracy in the structural dynamics solution, as well as smoothness in the string shape and fluid dynamics forces computed on the string. The STNMUM enables exact representation of the mesh rotation. The general-purpose NURBS mesh generation method makes it easier to deal with the complex geometry we have here. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small.

Dynamic term-by-term stabilized finite element formulation using orthogonal subgrid-scales for the incompressible Navier–Stokes problem

In this paper, we propose and analyze the stability and the dissipative structure of a new dynamic term-by-term stabilized finite element formulation for the Navier–Stokes problem that can be viewed as a variational multiscale (VMS) method under some assumptions. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual-based formulations, the introduction of two velocity subscale components. They represent the components of the convective and the pressure gradient terms, respectively, of the momentum equation that cannot be captured by the finite element mesh. A key idea of the proposed method is that the convective subscale is close to a solenoidal field and the pressure gradient subscale is close to a potential field. The method ensures stability in anisotropic space–time discretizations, which is proved using numerical analysis for a linearized problem and demonstrated in classical numerical tests. The work includes a detailed description of the proposed formulation and several numerical examples that serve to justify our claims.

Computational analysis of flow-driven string dynamics in a pump and residence time calculation

We present computational analysis of flow-driven string dynamics in a pump and the related residence time calculation. The objective in the study is to understand how the strings carried by a fluid interact with the pump surfaces, including the blades, and get stuck on or around those surfaces. The residence time calculations help us to have a simplified but quick understanding of the string behavior. The core computational method is the Space–Time Variational Multiscale (ST-VMS) method, and the other key methods are the ST Isogeometric Analysis (ST-IGA), ST Slip Interface (ST-SI) method, ST/NURBS Mesh Update Method (STNMUM), a general-purpose NURBS mesh generation method for complex geometries, and a one-way-dependence model for the string dynamics. The ST-IGA with NURBS basis functions in space is used in both fluid mechanics and string structural dynamics. The ST framework provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the turbulent nature of the unsteady flow, and the moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the pump geometry and increased accuracy in the flow solution. The IGA discretization also enables increased accuracy in the structural dynamics solution, as well as smoothness in the string shape and fluid dynamics forces computed on the string. The STNMUM enables exact representation of the mesh rotation. The general-purpose NURBS mesh generation method makes it easier to deal with the complex geometry. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small.

Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number

In this paper, we consider up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared using a high Reynolds number vortex dynamics simulation. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov–Galerkin (SUPG) method together with grad–div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulation of a high Reynolds number flow with vortical structures on relatively coarse grids are showcased, by focusing on a two-dimensional plane mixing-layer flow. Both Inf–Sup Stable (ISS) and Equal Order (EO) H1-conforming FE pairs are explored using a second-order semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies conducted, it is concluded that the SUPG method using EO–FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend the SUPG method by the higher order terms of the RB-VMS method.

Thermoelasticity at finite strains with weak and strong discontinuities

This paper presents a monolithic coupled formulation for finite strain thermoelasticity with weak and strong discontinuities. Thermal and mechanical properties can vary sharply across embedded interfaces where weak discontinuities are allowed to grow into strong discontinuities as the coupled nonlinear process evolves. Significant contributions in this work are: (i) variationally consistent coupling of multiple fields that have jumps across embedded interfaces, (ii) non-matching meshes with provision of different element types across interfaces, (iii) evolution of interfacial kinematics via embedding phenomenological models at the interfaces, and (iv) consistent linearization of the two-way coupled formulation leading to quadratic convergence of the method. The new Variational Multiscale Coupled Field (VMCF) formulation is developed by embedding Discontinuous Galerkin (DG) ideas in the Continuous Galerkin (CG) method within the context of the Variational Multiscale (VMS) framework. Starting from a thermomechanically coupled formulation over the elastic domain with Lagrange multipliers that couple fields along the interfaces, the Lagrange multipliers are eliminated by deriving analytical expressions for the multipliers via the interfacial fine-scale problems facilitated by the VMS framework. The derived terms for interfacial stabilization are a function of the residual of Euler–Lagrange equations along the interfaces. Stabilization tensors are functions of the evolving mechanical and thermal fields and are free of user-defined parameters. Several test cases are presented to illustrate the versatility and the range of applicability of the method.

Analysis of Robustness of Variational Multiscale Error Estimators for the Forward Propagation Study

Error estimation and control play a crucial role in the accuracy and reliability of numerical simulations. Variational multiscale (VMS) methods have emerged as effective techniques for error estimation in forward propagation studies. The robustness and accuracy of VMS error estimators under different conditions and parameters need to be thoroughly investigated. This research paper focuses on the analysis of the robustness of variational multiscale error estimators for forward propagation studies. The study explores the performance and limitations of VMS error estimators under various scenarios, including different mesh resolutions, material properties, and boundary conditions. The research aims to provide insights into the effectiveness and reliability of VMS error estimators, contributing to the improvement of forward propagation simulations. The analysis of robustness of variational multiscale error estimators for the forward propagation study addresses the important challenge of accurately assessing and quantifying the errors in computational models used for forward propagation. Variational multiscale error estimators have gained significant attention as they provide a reliable framework for error estimation in complex physical simulations. However, the robustness of these estimators, particularly in the context of forward propagation, remains an open question. To analyze the robustness of variational multiscale error estimators for the forward propagation study. We begin by formulating the problem and defining the mathematical framework for error estimation. Next, we investigate the performance and reliability of variational multiscale error estimators under different scenarios, including variations in model parameters, mesh resolution, and boundary conditions. We evaluate the accuracy and consistency of the estimators by comparing the estimated errors with reference solutions or known analytical solutions.
 We examine the influence of different factors, such as spatial and temporal discretization schemes, on the robustness of variational multiscale error estimators. We assess the sensitivity of the estimators to modeling assumptions and potential sources of error, such as numerical approximation and solution regularization. The effect of different physical phenomena, such as material nonlinearity or complex boundary conditions, on the performance of the estimators. The findings of this provide valuable insights into the robustness and limitations of variational multiscale error estimators for the forward propagation study. Understanding the performance characteristics and potential weaknesses of these estimators is crucial for accurately assessing the reliability and uncertainty of computational models used in forward propagation. The analysis of robustness contributes to the development of improved error estimation techniques and enhances the confidence in the results obtained from computational simulations. This focuses on the analysis of robustness of variational multiscale error estimators for the forward propagation study. By investigating their performance under different scenarios and evaluating their sensitivity to various factors, we provide a comprehensive understanding of the reliability and limitations of these estimators. The insights gained from this analysis have implications for improving the accuracy and confidence in computational models used for forward propagation in diverse fields, including engineering, physics, and computational sciences.

A novel characteristic variational multiscale FEM for incompressible natural convection problem with variable density

PurposeThe purpose of this paper is to propose a new method to solve the incompressible natural convection problem with variable density. The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods and to deal with high Rayleigh number for the natural convection problem.Design/methodology/approachThe paper introduces a novel characteristic variational multiscale (C-VMS) finite element method which combines advantages of both the characteristic and variational multiscale methods within a variational framework for solving the incompressible natural convection problem with variable density. The authors chose the conforming finite element pair (P2, P2, P1, P2) to approximate the density, velocity, pressure and temperature field.FindingsThe paper gives the stability analysis of the C-VMS method. Extensive two-dimensional/three-dimensional numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.Originality/valueExtensive 2D/3D numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.

Dynamic homogenization of resonant elastic metamaterials with space/time modulation

We present a variational multiscale strategy in the spirit of FE $$^2$$ method for simulating the real time evolution of resonant elastic metamaterials with space/time modulation. The implicit time discretization used guarantees the stability of the numerical solution, while the accuracy is quantified by direct comparison with the response obtained via direct numerical simulation. The results showcase a remarkable accuracy of the method, even when the dynamic material properties drastically change over length scales that are equal to the macroscopic mesh size and time scales that are of the same order of magnitude than the external excitations. We further propose a strategy to identify the band gap structure of these metamaterials based on their real time response. The methodology is verified for spatially periodic time invariant systems via classical unit cell analyses in the frequency domain.

Reduced order models for thermally coupled low Mach flows

In this paper we present a collection of techniques used to formulate a projection-based reduced order model (ROM) for zero Mach limit thermally coupled Navier–Stokes equations. The formulation derives from a standard proper orthogonal decomposition (POD) model reduction, and includes modifications to improve the drawbacks caused by the inherent non-linearity of the used Navier–Stokes equations: a hyper-ROM technique based on mesh coarsening; an implicit ROM subscales formulation based on a variational multi-scale (VMS) framework; and a Petrov–Galerkin projection necessary in the case of non-symmetric terms. At the end of the article, we test the proposed ROM formulation using 2D and 3D versions of the same example: a differentially heated cavity.

Non-transferred Arc Torch Simulation by a Non-equilibrium Plasma Laminar-to-Turbulent Flow Model

Non-transferred arc torches are at the core of diverse industrial applications, particularly plasma spray. The flow in these torches transitions from laminar inside the torch to turbulent in the emerging jet. The interaction of the plasma with the processing gas leads to significant deviations from local thermodynamic equilibrium (LTE) far from the arc core. The flow from a non-transferred arc plasma spray torch is simulated using a non-LTE (NLTE) plasma flow model solved by variational multiscale (VMS) and nonlinear VMS (VMSn) methods, which are suitable for unified laminar and turbulent flow simulations. Non-plasma turbulent jet simulations indicate that the VMSn method produces results comparable to those by the dynamic Smagorinsky method, often considered the workhorse for turbulent incompressible flow simulations. VMS and VMSn approaches are applied to the simulation of incompressible, compressible, and NLTE plasma flows in non-transferred arc torch operating at representative conditions found in plasma spray processes. The NLTE plasma flow simulations reproduce the dynamics of the arc inside the torch together with the evolution of turbulence in the produced plasma jet in a cohesive manner. However, the similarity of results by both methods indicates the need for numerical resolution significantly higher than what is commonly afforded in arc torch simulations.

Tire aerodynamics with actual tire geometry, road contact and tire deformation

Tire aerodynamics with actual tire geometry, road contact and tire deformation pose tough computational challenges. The challenges include (1) the complexity of an actual tire geometry with longitudinal and transverse grooves, (2) the spin of the tire, (3) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flow-domain topology change created by the road contact and tire deformation, and (4) the turbulent nature of the flow. A new space–time (ST) computational method, “ST-SI-TC-IGA,” is enabling us to address these challenges. The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). The VMS feature of the ST-VMS addresses the challenge created by the turbulent nature of the flow, the moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces, and the higher-order accuracy of the ST framework strengthens both features. The ST-SI enables moving-mesh computation with the tire spinning. The mesh covering the tire spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-TC enables moving-mesh computation even with the TC created by the contact between the tire and the road. It deals with the contact while maintaining high-resolution flow representation near the tire. Integration of the ST-SI and ST-TC enables high-resolution representation even though parts of the SI are coinciding with the tire and road surfaces. It also enables dealing with the tire–road contact location change and contact sliding. By integrating the ST-IGA with the ST-SI and ST-TC, in addition to having a more accurate representation of the tire geometry and increased accuracy in the flow solution, the element density in the tire grooves and in the narrow spaces near the contact areas is kept at a reasonable level. We present computations with the ST-SI-TC-IGA and two models of flow around a rotating tire with road contact and prescribed deformation. One is a simple 2D model for verification purposes, and one is a 3D model with an actual tire geometry and a deformation pattern provided by the tire company. The computations show the effectiveness of the ST-SI-TC-IGA in tire aerodynamics.

Separation delay through contoured transverse grooves on a 2D boat-tailed bluff body: Effects on drag reduction and wake flow features

Abstract The effectiveness of properly contoured transverse grooves in delaying the flow separation occurring on a two-dimensional boat-tailed bluff body is assessed through numerical simulations. The body has a cross-section with a 3:1 elliptical forebody and a rectangular main part followed by a circular-arc boat tail. Three-dimensional Variational Multiscale Large Eddy Simulations are carried out at R e = D u ∞ ∕ ν = 9 . 6 × 1 0 4 , using a mixed finite-volume/finite-element method. The introduction of one contoured groove on each of the boat-tail lateral surfaces produces a significant delay of flow separation and a consequent increase of the base pressure, with a global drag reduction of the order of 9 . 7 % . The wake dynamical structure remains qualitatively similar to the one typical of blunt-based two-dimensional bodies, with quantitative variations that are consistent with the reduction in wake width caused by boat tailing and by the grooves. The introduction of the grooves leads also to a regularization of the vortex shedding downstream of the body, which is more correlated in the spanwise direction. Finally, a few supplementary simulations show that the effect of the grooves is also robust to the variation of the geometrical parameters defining their location and shape.

Time-accurate calculation and bifurcation analysis of the incompressible flow over a square cavity using variational multiscale modeling

A thorough variational multiscale (VMS) modeling of the Navier–Stokes equations is used to compute numerical solutions of the incompressible flow over an open cavity. This case features several competing instabilities, and is highly challenging for VMS methods with regard to frequency and pattern selection, because of the non-normality of the linearized Navier–Stokes operator. The relevance of the approach is thus carefully assessed by comparing to direct numerical simulation (DNS) data benchmarked at several Reynolds numbers, and highly accurate time advancing methods are shown to predict relevant evolutions of the transient and saturated solutions. The VMS reduces substantially the computational cost, by ∼35% (resp. ∼60%) in terms of CPU time using a semi-implicit discretization scheme based on backward differentiation formula (resp. the implicit Crank–Nicholson scheme), and by ∼80% in terms of memory requirement. Eventually, the highly efficient semi-implicit VMS numerical framework is used to unravel the onset of the flow oscillations and the selection of the limit cycle frequency, that happens to involve a subcritical Neimark–Sacker bifurcation.

A variational multiscale discontinuous Galerkin formulation for both implicit and explicit dynamic modeling of interfacial fracture

A stabilized Discontinuous Galerkin method is presented for modeling interfacial debonding under dynamic conditions. The method’s distinctive feature is its treatment of the inelastic gap at the interface as an internal variable that is evolved through a traction–separation constitutive model, enabling initially perfect adhesion to be captured. Numerical stability is derived using Variational Multiscale ideas, and comparisons are drawn with both the intrinsic cohesive zone method and hybrid Discontinuous Galerkin-cohesive zone method to expose subtle differences. The proposed formulation is applied to model the nonlinear response of a composite unit cell under impact loading. Both explicit and implicit time integration schemes provide similar accuracy and stability, and the critical time step for explicit analyses is only slightly reduced below the value for the continuous Galerkin mesh without an interface. Further studies of mesh refinement and adjustments to the initial elastic interface stiffness in the reference cohesive zone model reveal the improved accuracy of the Variational Multiscale Discontinuous Galerkin method on coarser grids.

Numerical modeling of local capillary effects in porous media as a pressure discontinuity acting on the interface of a transient bi-fluid flow

Transient flows through porous media can be controlled by local capillary forces. In an attempt to ease the representation of these complex multi-scale flows, this article presents a new numerical approach to account for these local forces, viewed as a global pressure discontinuity acting in bi-fluid flows through smeared-out porous media. A finite element discretization of the Darcy’s equations is considered and a pressure enriched space is locally introduced at the fluid interface in order to capture the pressure discontinuity. Then, a Variational Multiscale Stabilization (VMS) method is selected to take into account the subgrid effects on the finite element solution and hence ensure the consistency of the finite element formulation. The fluid front is represented by a level set function, convected with the fluid velocity thanks to a finite element scheme stabilized with a Streamline-Upwind/Petrov-Galerkin (SUPG) method. Both convergence and implementation are first validated with the Method of Manufactured Solution (MMS) and the model shows a good convergence. Second, a comparison with experimental measurements in the case of capillary wicking of water into carbon reinforcements shows a very good correlation between experimental and numerical results.

On the robustness of variational multiscale error estimators for the forward propagation of uncertainty

The numerical simulation of physical phenomena and engineering problems can be affected by numerical errors and various types of uncertainties. Characterizing the former in computational frameworks involving system parameter uncertainties becomes a key issue. In this work, we study the behavior of new variational multiscale (VMS) error estimators for the propagation of parametric uncertainties in a Convection–Diffusion–Reaction (CDR) problem. A sensitivity analysis is performed to assess the performance of the error estimator with respect to the mesh discretization and physical parameters (here, the viscosity value and advection velocity). Three different manufactured analytical solutions are considered as benchmarking tests. Next, the performance of the VMS error estimators is evaluated for the CDR problem with uncertain input parameters. For this purpose, two probabilistic models are constructed for the viscosity and advection direction, and the uncertainties are propagated using a polynomial chaos expansion approach. A convergence analysis is specifically carried out for different configurations, corresponding to regimes where the CDR operator is either smooth or non-smooth. An assessment of the proposed error estimator is finally conducted for the three tests, considering both the viscous- and convection-dominated regimes.

Variational Multiscale Modeling of Lithium Batteries

Batteries are required to have higher energy and power density to meet the requirement of future electric vehicles and other power supply applications. To ensure the safety and durability of the high energy lithium batteries, the numerical study is the key to understanding and design of the next generation batteries. More importantly, the modeling and simulation study of lithium batteries help us understand the underlying causes of the electrochemical, electrical, thermal, and mechanical performance of a lithium battery under various operating conditions. However, the traditional modeling method of lithium batteries assumes the homogeneous structure and reaction to reduce the complexity of the model. The heterogeneity of the battery electrodes significantly impacts the performance of a battery. Due to the complex microstructure of the battery electrodes, it requires ultra-fine meshes to model the various length scales of the electrode structure, which requires very high computing power. This high computational cost is not practical for the numerical optimization design and long-term cycling simulations. Therefore, a variational multiscale method (VMM) is the key to model a lithium battery with considering the heterogeneity. The proposed VMM model reduces the computational cost of a realistic 3D battery model without compromise the accuracy of the heterogeneity. The VMM splits the battery model into macro-scale and micro-scale. By applying the micro-green function, which was firstly developed by Hughes [1] and then extended to battery modeling by us [2], as illustrated in Fig. 1, the micro-scale problem is analytically solved through the projection based on the macro-scale solution to obtain more accurate results. By doing so, we can significantly increase the computational efficiency and accuracy of conducting complex battery multi-scale simulations. With this method, we can simulate battery degradations in the realistic microstructure over hundreds of charge and discharge cycles with a reasonable computational cost. With the understanding of the battery behavior over long-term cycling, the battery electrodes can be redesigned to obtain longer lifetime and better performance through the numerical optimization for energy storage applications. [1] T.J. Hughes, G. Sangalli, Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM Journal on Numerical Analysis, 2007. 45(2): p. 539-557. [2] L. Liu, M. Moradi, Towards a More Realistic Battery Model: Variational Multiscale Modeling of Li-Ion Batteries, ECS Transactions, 2017. 77(11): p. 273-291 Figure 1